Determination of lattice parameters with the aid of a computer

Mueller, M. H.; Heaton, L.; Miller, K. T.

doi:10.1107/s0365110x60002004

Cited by 130 publications

(37 citation statements)

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“…Forty-one resolved lines in the 20 region from 102°-165 ° were used to calculate the lattice parameters (Cu K0q = 1.54050 A, Cu K0~2 = 1.54434 A) with the computer program of Mueller, Heaton & Miller (1960). These results and other physical constants for YzSiBeEO7 are given in Table 1.…”

Section: Crystal Datamentioning

confidence: 99%

Crystal structure of Y₂SiBe₂O₇

Bartram¹

1969

Acta Crystallogr Sect B

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Diyttrium silicon beryllate, Y2SiBe207, is tetragonal with a = 7.283 _+ 0-002, c = 4.755 _+ 0.001 /~, Z= 2. Refinement of its crystal structure in space group P~21m by full-matrix least-squares calculations gave R=0.096. Yttrium atoms lie within distorted square oxygen antiprisms and silicon atoms in isolated SiO4 tetrahedra. Beryllium atoms occupy distorted tetrahedra linked at one corner to form double Be207 pyramids oriented upward or downward relative to the e axis. Y2SiBezOv is isostructural with silicate minerals of the melilite family, such as CazSiAI207 and CazMgSi207. The M207 group is capable of accepting various small ions of different valence, giving this structural arrangement a remarkable versatility.

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Section: Crystal Datamentioning

confidence: 99%

Crystal structure of Y₂SiBe₂O₇

Bartram¹

1969

Acta Crystallogr Sect B

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“…The symmetry and the lattice parameter were determined from a powder film taken at room temperature with a Norelco camera of 114.6 cm in diameter and Cr Ke radiation. The lattice parameter was calculated by the leastsquares refinement program of Mueller, Heaton & Miller (1960). A comparison of observed and calculated interplanar spacings and observed intensities is given in Table 1.…”

Section: Crystal Datamentioning

confidence: 99%

The crystal structure of the high-pressure phase CaB₂O₄(IV), and polymorphism in CaB₂O₄

Marezio¹,

Remeika²,

Dernier³

1969

Acta Crystallogr Sect B

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CaB204(IV) is a high pressure phase of calcium metaborate. Single crystals were grown at 30 kbar and 900°C. They were found to be cubic with a=9.008 + 0.001 A,, Z= 12, de= 3.426 g.cm-3 and space group symmetry Pa3.The structure has been determined by a three-dimensional Fourier synthesis. The positional and isotropic thermal parameters have been refined by the least-squares method based on 267 structure factors. The final conventional R value is 0.033. All the boron atoms are tetrahedrally coordinated. The mean B-O distance is 1.480/~. There are two independent Ca atoms. Ca(I) is surrounded by a 12-oxygen polyhedron with two sets of Ca-O distances of 2-671 A, and 2.785 A,. This is the first observation of coordination twelve for calcium. Ca(2) is also surrounded by twelve oxygens, but the polyhedron formed by these oxygen atoms is very irregular. There are four sets of Ca-O distances: 2.385, 2.483, 2.598 and 3.142 A,. The coordination number of the oxygen atoms is five; each one is surrounded by one Ca(l), two Ca(2), and two boron atoms.Calcium metaborate can crystallize with four different structures by varying the pressure at which the synthesis is carried out. The common trend among these structures is an increase of the cation coordination with increasing pressure. By analogy many new phases of borates and polyborates could be synthesized under pressure.

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“…To define the problem let vj, given by (Mueller, Heaton & Miller, 1960;Plackett, 1960;Caron & Donohue, 1961;Kempter & Vogel, 1962;Mozzi & Newell, 1962;Evans, Appleman & Handwerker, 1963). Two important assumptions are involved in the least-squares method: (1) the correct assignment of hjlcflj is assumed for each observed reflection and (2) the residuals, Vh are assumed to follow the Gaussian law of error (Fig.…”

Section: Comparison With Least-squares Methodsmentioning

confidence: 99%

The determination of axial ratios from powder diffraction patterns

Frevel¹

1964

Acta Crystallogr

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Ace urate axial ratios of anisotropic crystalline phases can be obtained from precision powder data by computing exhaustively the axial ratios from pairs of closely spaced, non-overlapping reflections. The method has been applied successfully to the tetragonal, hexagonal, orthorhombic, and monoclinic systems.The determination of axial ratios with the reflecting goniometer was developed into a precise technique by mineralogists of the nineteenth century. Compendia such as Dana's System of Mineralogy or Groth's Chemische Krystallographie are replete with this type of morphological information which has been of A C 17 --59 908 AXIAL RATIOS FROM POWDER DIFFRACTION PATTERNS immense value to X-ray crystallographers. During the past three decades, mineralogists have shifted from the determination of axial ratios by optical goniometry to the determination of unit-cell dimensions by X-ray diffraction. Single-crystal methods, in general, have proved vastly superior to the powder methods in establishing the correct unit cell of a crystalline phase. However, in many cases, the accuracy of these cell constants is not sufficiently high to yield axial ratios comparable in accuracy to those determined by the two-circle goniometer. The reason for this limited accuracy can be attributed largely to the general use of small-radius cameras for rotation or Weissenberg photographs. Modern powder methods, on the other hand, are capable of yielding axial ratios of greater accuracy than those obtained by the best morphological measurements (Frondel, 1962). There is an abundance of published data on precision measurements of lattice parameters of polycrystalline phases (Klug& Alexander, 1954;Edmunds, Lipson & Steeple, 1955;Azaroff & Buerger, 1957; Parrish & Wilson, 1959; I.U.Cr. Stockholm meeting, 1959); unfortunately, however, the substances investigated pertain to relatively simple structures for which unequivocally indexed back-reflections can be registered. For anisotropic substances with cell dimensions exceeding 6 ~, it is rather unusual to observe in the back-reflection region unambiguous pinacoid reflections of sufficient intensity for the reliable determination of lattice parameters. To circumvent this limitation of the powder method it has been found advantageous to compute exhaustively the axial ratios from pairs of closely spaced, nonoverlapping reflections. TheoryThe ratio of two interplanar spacings, dm=d (hmkmlm) and dn = d(hnlcnln), can be determined with a minimum error when these spacings form a clearly resolved 'doublet' because as 0n approaches 0~ (without overlapping) the principal systematic errors cancel out; e.g. film shrinkage, effective camera radius, absorption correction, beam divergence, and refraction correction at low 0. For an orthorhombic crystal, the square of this ratio is given by the expression where rl =a/b, re = c/b, the cell edges follow the convention 0 < c < a

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Determination of lattice parameters with the aid of a computer

Cited by 130 publications

References 1 publication

Crystal structure of Y₂SiBe₂O₇

Crystal structure of Y₂SiBe₂O₇

The crystal structure of the high-pressure phase CaB₂O₄(IV), and polymorphism in CaB₂O₄

The determination of axial ratios from powder diffraction patterns

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Determination of lattice parameters with the aid of a computer

Cited by 130 publications

References 1 publication

Crystal structure of Y2SiBe2O7

Crystal structure of Y2SiBe2O7

The crystal structure of the high-pressure phase CaB2O4(IV), and polymorphism in CaB2O4

The determination of axial ratios from powder diffraction patterns

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Crystal structure of Y₂SiBe₂O₇

Crystal structure of Y₂SiBe₂O₇

The crystal structure of the high-pressure phase CaB₂O₄(IV), and polymorphism in CaB₂O₄